7. Boundary Conditions Summary

In this lecture, we proceed forward with the bottom boundary conditions also called as, BBC. Let us say the bottom is described as z = - h of x. If we consider 0 at the free surface, the origin is located at still water level. From the boundary condition, we see the bottom is fixed for u · n = 0.

The Bottom Boundary Condition (BBC) is crucial in hydraulic engineering. We start by defining the riverbed or seabed, described mathematically as z = -h(x). This means that if we set the water's surface at 0 (which is our reference point), the depth changes along the x-axis according to the function h(x). Since the boundary at this depth is fixed (the water cannot pass through the bed), the velocity component in the vertical direction (denoted as u · n) must equal 0. This indicates that at the seabed, there's no motion of the water vertically, confirming that the seabed acts as a wall, blocking water flow.

Imagine standing at the edge of a swimming pool. The pool’s bottom represents the seabed. When you jump in, the water is prevented from going any lower than the bottom of the pool (the fixed boundary). Just like the bottom of the pool holds water up and keeps it contained, the seabed does the same function in nature, which is why we must model how water behaves at these boundaries.

If we assume a case of a horizontal bottom, we know that dh/dx is going to be 0. Therefore, this means w = 0. Thus, we have proved that the boundary condition for w = 0 using the bottom boundary condition.

When we consider a horizontal bottom, the change in height with respect to the horizontal distance (dh/dx) is zero. This mathematically means that there’s no slope; the bottom of the water body is flat. Because of this flatness, the vertical velocity component (w) also equates to zero. Thus, the water particles at the bottom don’t move up or down, maintaining a static position against the horizontal seabed.

Think of a still lake on a calm day where the bottom is flat, just like a level surface on a table. If you were to drop a ball, it would come to rest on the flat bottom and wouldn’t flow up or down. The same principle applies here; the flat ocean floor keeps the water stable above it.

In the case of a sloping bottom, we can write w/u = -dh/dx. This indicates that as the bottom slopes, the vertical velocity of the water must correspondingly adjust to maintain flow conditions.

For a sloping seabed, the relationship between the vertical (w) and horizontal (u) water velocities becomes crucial for maintaining continuity in flow. The equation w/u = -dh/dx signifies that the vertical velocity w is inversely related to how steep the slope (dh/dx) is. This means that a steeper slope will result in a faster vertical water movement.

Imagine sliding down a playground slide. The steeper the slide, the quicker you go; similarly, in water, if the riverbed slopes steeply, the water races faster downwards along that incline, changing how it moves vertically.

We have another condition called the dynamic free surface boundary condition. The displacement of the upper boundary in free surface problems is not known, making it essential to prescribe another boundary condition to analyze pressure distributions.

Dynamic free surfaces are crucial when analyzing fluid behavior because they can change shape under varying conditions (such as waves). Unlike fixed boundaries which hold a known position and can support pressure changes, free surfaces cannot support pressure variations and will instead deform. This necessitates another boundary condition to describe how pressure varies across the free surface, ensuring accurate predictions of how liquid behaves.

Picture a water balloon. When you press on it, the shape changes, and so does the pressure. Similarly, the surface of the ocean shifts with wind and waves, making it dynamic and requiring us to understand this behavior to predict how water will flow and respond to various forces.

The requirement for the dynamic free surface boundary condition is that pressure on the free surface must be uniform along the waveform. To derive this, we turn to the unsteady Bernoulli’s equation.

For flowing fluids like water in waves, the dynamic boundary condition requires that pressure remains uniform at the surface across different points. This means the pressure does not vary along the surface waveform, which can be adapted from Bernoulli’s equation. The uniformity of pressure is necessary for the stability of waves and other phenomena at the surface.

Think of a long line of kids on a swing set, all swinging back and forth together. If one kid swings higher or lower than the rest, it disrupts the flow of the entire line. Just like the kids, for waves to be stable, pressure must be consistent across the surface of the water.

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